High-frequency Green's function for a rectangular array of dipoles with weakly varying tapered excitation

The array Green's function (AGF) is the basic building block for the full-wave analysis of planar phased array antennas. Its representation in terms of element-by-element summation over the individual dipole radiations can be replaced by a more efficient global representation constructed via Poisson summation. The resulting Poisson-transformed integrals can be interpreted as the radiation from continuous equivalent Floquet wave (FW)-matched source distributions extending over the array aperture. Applying high-frequency asymptotics to each FW-matched array aperture casts the AGF in the format of a generalized geometrical theory of diffraction (GTD) which includes conical wavefront edge diffracted rays as well as spherical wavefront vertex diffracted rays. In this paper, the results are extended to accommodate tapered illumination, which also includes dipole amplitudes tending to zero at the edges. This extension is carried out by a direct Poisson-transformed asymptotic evaluation of the strip array GF, with inclusion of asymptotically subdominant "slope" edge and vertex diffracted fields, in addition to the dominant edge and vertex diffracted fields for appreciable edge illumination. Numerical results are presented for illustration.


I. INTRODUCTION
The array Green's function (AGF) is the basic building block for the full-wave analysis of planar phased array antennas.Its representation in terms of elementby-element summation over the individual dipole radiations can be replaced by a more efficient global representation constructed via Poisson summation.The resulting Poisson-transformed integrals can be interpreted as the radiation from continuous equivalent Floquet wave (FW)-matched source distributions extending over the array aperture [l], [2].Applying high-frequency asymptotics to each FW-matched array aperture casts the AGF in the format of a generalized Geometrical Theory of Diffraction (GTD) which includes conical wavefront edge diffracted rays as well as spherical wavefront vertex diffracted rays.In this paper, the results in [I], valid for equiamplitude excitation, are extended to accommodate tapered illumination, which also includes dipole amplitudes tending to zero at the edges.This extension, which has been performed in [3] with a numerical technique based on the discrete Fourier transform (DFT), is herein carried out by a direct Poisson-transformed asymptotic evaluation of the striparray GF with inclusion of asymptotically subdominant "slope" ed e and vertex diffracted fields, in addition to the dominant edge and vertex difFractecffields for appreciable edge illumination.Numerical results are presented for illustration.

FORMULATION
Consider a rectangular periodic array of NI x Nz linearly phased dipoles located in the zl,zz-plane (Fig. la), with interelement spatial period along the 21 and zz directions given by dl and d2, and the interelement phase gradient by 71 and 72, respectively.

HIGH-FREQUENCY SOLUTION
Henceforth, we assume [legitimately for actual tapering functions for large arrays) that f;(z,) varies slowly with respect to the wavelength A. For such weak variation, and since fi(zj) is positive in the domain z; E (O,L,), its spectrum f ; ( k l i ) is localized around k l j = 0, thereby enhancing contributions to l i ( k l i ) from k,j = ksiu.Thus, adiabatic methods can be applied, based on perturbation about fj(zi) = const.. Consequently, the integral in (1) which defines A is dominated asymptotically by: a) one ( k z l r k,z) saddle-point ( k : l , k&) that satisfies ( d / d k z i ) g ( k z l , k,z) = 0, yielding the vertex diffracted field; b)"quasi poles" at k,i = k,ju that describe the same phenomenology and localization property_ as the spectral poles for the semi-infinite array [l], yielding FWs; c)critical points at k i l and kZ2 which annul ( d / d k , l ) g ( k , l , k z z , ) and  ( d / d k z z ) g ( k i l q , k,z) respectively, and lead to diffracted fields from edge 2 (located at z1 = 0) and edge 1 (at z 2 = 0), respectively.Diffraction from the other two edges can be found similarly by including the appropriate phase reference in the second term in the right side of (2).
Floquet Wave Contributions.Inserting (2) into (l), the contributions due to the critical points at ( k z l , k,z) = (kZlq, kzzp) are found by expanding the exponent of the integrand in Taylor series in a neighborhood of ( k z l q , kzzp) (see [4]).Retaining only the dominant asymptotic term of the remainder which applies for observation points away from the array surface one finals where z,,, = r, -ykrru/kgpqr and kgpq = 4 -(with branches chosen according to (1)) is real for propagating FW, and U, , = U ( Z ~~, ) U ( ~~~, ) U ( L ~ zl,,)U(Lzzz,,) with U ( z ) = 1 or 0 if z > 0 or z < 0 respectively.Criteria for the asymptotic validity of the expansion obtained in (3) will be given elsewhere.In (3), AgW is the pqth FW for eqniamplitude excitation [l], which is multiplied in (3) by the tapering function j(z~,,, zzPq) evaluated at the footprint ( z l p p r zzpq) of the pqth FW.The stationary phase evaluation of the radiation integral associated with each p, qth equivalent FW-matched aperture distribution would provide the same result (see 2D case in [5])since (zip,, zzPq) is the stationary phase point of the p, qth spatial radiation integral.The function up,, is unity or zero for ( z l p q , is inside or outside the finite array dimensions.The discontinuity of the truncated FW at the Shadow boundary (SB) plane angle defined by zip, = 0 is restored by the diffracted field that arises from the saddle point evaluation of (1).

FW-Induced diffracted contributions.
Here, we show only the final result, derived asymptotically in [4], for the diffracted field arising from the truncation at z1 E [O,LI] (edge 1) associated with the critical spectral points (k~l~o,k~z).The total propagating diffracted field arising from edge 1 is thus represented as Ad>' = C, A$'lJ,OIJ,L1 where in which Bz and its derivative B;(kZz) are defined in the text after (5), F ( z ) is the standard UTD transition function, F S ( z ) = Zjz[l -F(z)] is the slope UTD transition function with argument S:,pq (defined in [4]) which vanishes at the SB planes, and U," = U(@: -PI), where 0 1 is the observation angle (see Fig. la) (similarly for U,"l).A$' is the q-th conical wave decaying along p1 and , G' ?f = 01 = cm-*(kzl,,/k) locates the shadow boundary cone (SBC) centered at the vertex, which truncates the domain of existence of the el-edge diffracted waves.In (4) the tapering function f1 is evaluated at the diffraction points zf, = z1 -ykZ1,,/k& with k;, = , / -.
The discontinuity across the SBC of the edgediffracted contribution is repaired by the diffraction from the vertex (0,O) of the array.Analogous diffracted fields arise from the other edges.

Vertex diffracted contributions.
Near the vertex at (qrz2) = (O,O), for example, the q-edge and zz-edge planar FW-shadow boundaries interact with the vertexinduced conical SBCs with symmetry axes z1 and Z Z , respectively, that arise form the truncation of the corresponding edge diffracted fields.The confluence of these four SBs near the vertex defines the asymptotics pertaining to the vertex diffracted field A",'(r), obtained by the following steps.First, I, in the (2) is expanded asymptotically (integrating by parts up to the second order) as domain 21 E [0, Lt], z~ E [O, Lz] and zero elsewhere; here, L1 = ( N I -1)dl and LZ = (Nz-l)dz.The electromagnetic vector field at any observation point r = zlil +zziz+ yy can be derived from the vector potential A(r) = joA(r) by summing over the individual (nl, nz) dipole radiations fl(n,d,)fi(nldl)e-l(l~"ldl+~~"~da) exp(-jk~,.,)/(4nE~"~)where, R,,,, = lrnldlilnzdzizl.Employing the (kZ1,krz) spectral Fourier representation of the free space Green's function as shown in [l], yields with I;(kz;) Fig. 1. (a)Geometry of the rectangular array, pI = m, pz = m.( b ) ! , , component of the electric field radiated by a rectangular array with excitation function f(zl, z2) = ni sin(mi/Li) , at a distance R = 1OX from the vertex.